Method and sheet like sensor for measuring stress distribution

ABSTRACT

A method and sheet-like sensor for measuring stress distribution including a grid of members which change in resistance when subjected to strain, the members intersecting at internal nodes and intersecting at boundary nodes at the periphery of the grid defining a plurality of legs. An analyzer is electrically connected only to the boundary nodes and configured to calculate any change in resistance in all of the legs based solely on the measured resistance of the legs between the boundary nodes.

RELATED APPLICATIONS

This application claims priority of U.S. Provisional application serialNo. 60/373,096 filed Apr. 16, 2002 and U.S. Provisional applicationserial No. 60/373,058 filed Apr. 16, 2002.

FIELD OF THE INVENTION

This invention relates to a sheet-like sensor for, inter alia, measuringstress distribution experienced by structural components and to a methodfor measuring resistances in a network or grid of resistances.

BACKGROUND OF THE INVENTION

It is known in the art that certain materials exhibit a change inelectrical resistance as a function of strain experienced by a material.A grid of members (e.g., copper wires) which change in resistance as afunction of strain can be constructed and bonded to or integrated with astructural element (e.g., an aircraft wing) to detect the stressesexperienced by the structural element. But, electrical connections mustbe made to each node of the grid. For large systems with many nodes, thesheer number of electrical connections becomes unwieldy as do thecomputations required to measure the change in resistance of all thelegs between the nodes.

U.S. Pat. No. 5,650,570, incorporated herein by this reference,discloses a sheet-like sensor with amorphous iron-based alloy memberswoven into glass cloth layers separated by an insulating sheet andcovered by synthetic rubber sheets. The members of the first cloth layerrun parallel to each other and the members of the second cloth layer runparallel to each other but perpendicular to the members of the firstcloth layer. One end of all the members of the first cloth layer areelectrically connected to a first scanner and the other end of all ofthe members of the first cloth layer are electrically connected to afirst impedance analyzer. One end of all of the members of the secondcloth layer are electrically connected to a second scanner and the otherend of all of the members of the second cloth layer are electricallyconnected to a second impedance analyzer. In this way, the change inresistance along the length of any member due to strain can be measuredand the strain computed.

Unfortunately, the specific location of the strain experienced by thesensor cannot be detected. The same is true if a member fails: thesensor cannot identify the specific location of a failure. Moreover, themaximum strain that can be computed is limited by the failure strain ofthe ferromagnetic elements used which is between 0.2% and 0.4%. Finally,the method disclosed in the '570 patent cannot accurately predict thestress distribution of a structural component since it only provides anestimate of where a force or pressure is applied.

SUMMARY OF THE INVENTION

It is therefore an object of this invention to provide improvedsheet-like sensor for measuring stress distribution.

It is a further object of this invention to provide such a sensor whichreduces the number of electrical connections required to filly analyzethe stress experienced by a structural component.

It is a further object of this invention to provide such a sensor inwhich no electrical connections are required internal to the sensor tofully analyze the full stress distribution.

It is a further object of this invention to provide such a sensor whichis capable of detecting the specific location of the strains experiencedby the sensor.

It is a further object of this invention to provide such a sensor whichis capable of identifying the specific location of a failure.

It is a further object of this invention to provide such a sensor whichis able to measure strains of a higher magnitude.

It is a further object of this invention to provide such a sensor whichcan fully predict stress distribution.

It is a further object of this invention to provide a method ofdetermining all of the impedances of a grid of leg impedances.

It is a further object of this invention to provide such a method usefulin connection with a sheet-like sensor or in connection with analyzersof other electrical circuits.

This invention results from the realization that a better, lesscumbersome, more accurate, and more useful sheet-like sensor is effectedby arranging members which change resistance as a function of strain asa grid forming legs between both internal and external nodes but onlyconnecting the resistance measurement analyzer to the boundary nodes andthen determining all of the leg resistances based on the measuredresistances of the legs between the boundary nodes using an iterativealgorithm. In this way, the electrical interconnections between theanalyzer and the internal nodes of the grid are eliminated thusseriously reducing the number of electrical interconnections required.Moreover, the specific location of any strains experienced by the sensorcan be more accurately detected, the specific location of any failurecan be identified, and full stress distribution of a structural memberor component underlying the sensor can be predicted. In addition, byusing pseudoelastic shape memory alloy material instead of ferromagneticmaterials, strains of a higher magnitude can be measured. This inventionalso results from the realization that the algorithm used in connectionwith the analyzer of the sheet-like sensor can be used in otherenvironments, e.g., for evaluating electrical circuits.

This invention features a sheet-like sensor for measuring stressdistribution typically comprising a grid of members which change inresistance when subjected to strain, the members intersecting atinternal nodes and intersecting at boundary nodes at the periphery ofthe grid defining a plurality of legs. An analyzer is electricallyconnected only to the boundary nodes and configured to calculate anychange in resistance in all of the legs based solely on the measuredresistance of the legs between the boundary nodes.

In one example, the members are copper wires. In another example, thewires are made of pseudoelastic shape memory alloy material. The grid ofmembers may be encapsulated in an encapsulation material such as Kapton.In this way, the analyzer can be formed as a circuit integral with theencapsulation material. The grid may be in the shape of a polygon, e.g.,a rectangle or a square. Other shapes and designs, however, are possible

Typically, the analyzer is configured to measure the resistances of thelegs between the boundary nodes, to estimate the resistances of all ofthe legs, calculate the resistances of all of the legs based on themeasured resistances of the legs between the boundary nodes and theestimated resistances of all of the legs, and to compare the calculatedresistances of the legs between the boundary nodes with the measuredresistances of the legs between the boundary nodes. Based on thecomparison, a re-estimate of the resistances of all of the legs is made.Then, iterations of these steps are performed until the measuredresistances of the legs between the boundary nodes converge to thecalculated resistances of the legs between the boundary nodes to thusaccurately determine the resistances of the legs between or connected tothe internal nodes.

In one example, the analyzer is further configured to calculate thestrain experience by each leg. Also, the analyzer may be furtherconfigured to identify any leg which has failed based on a very highdetermined resistance. Typically, the initial estimate is based on themeasured resistances, e.g., the initial estimate is set to the mean ofthe measured resistances. Also, relaxation techniques may be used.

A sheet-like sensor for measuring stress distribution in accordance withthis invention typically includes a grid of members which change inresistance when subjected to strain, the members intersecting atinternal nodes and intersecting at boundary nodes at the periphery ofthe grid defining a plurality of legs. An analyzer is connected only tothe boundary nodes. In the preferred embodiment, the analyzer isconfigured to measure the resistances of the legs between the boundarynodes and estimate the resistances of all of the legs, calculate theresistances of all of the legs based on the measured resistances of thelegs between the boundary nodes and the estimated resistances of all ofthe legs. The calculated resistances of the legs between the boundarynodes is compared with the measured resistances of the legs between theboundary nodes. Based on the comparison, a re-estimate of theresistances of all of the legs is made, and iterations continue untilthe measured resistances of the legs between the boundary nodes convergeto the calculated resistances of the legs between the boundary nodes. Inthis way, the resistances of the legs between or connected to theinternal nodes is accurately determined.

This invention also features a sensor system or method for a gridincluding internal nodes and boundary nodes at the periphery of the griddefining a plurality of legs in which a characteristic of the legsbetween the boundary nodes is measured, the same characteristic of allof the legs is estimated, and the same characteristic of all of the legsis calculated based on the measured characteristic of the legs betweenthe boundary nodes and the estimated characteristic of all of the legs.Next, a comparison is made between the calculated characteristic of thelegs between the boundary nodes and the measured characteristic of thelegs between the boundary nodes. Based on the comparison, thecharacteristic of all of the legs is again estimated, and iterationscontinue until the measured characteristic of the legs between theboundary nodes converge to the calculated characteristic of the legsbetween the boundary nodes. In one example, the members change inresistance when subjected to strain and the characteristic analyzed isresistance which varies as a function of strain. In one example, thecharacteristics are complex impedences.

One exemplary method for determining impedances in a grid of legimpedances in accordance with this invention includes: a) measuring theresistances of the legs between the boundary nodes, b) estimating theresistances of all of the legs, c) calculating the resistances of all ofthe legs based on the measured resistances of the legs between theboundary nodes and the estimated resistances of all of the legs, d)comparing the calculated resistances of the legs between the boundarynodes with the measured resistances of the legs between the boundarynodes, e) based on the comparison, re-estimating the resistances of allof the legs, and f) iteratively repeating steps c)-e) until the measuredresistances of the legs between the boundary nodes converge to thecalculated resistances of the legs between the boundary nodes to thusaccurately determine the resistance of the legs between or connected tothe internal nodes. Further included may the steps of calculating thestrain experience by each leg, and identifying any leg which has failedbased on a very high determined resistance. Typically, the estimate ofstep b) is based on the step a), (e.g., the estimate is set to the meanof the measured resistances). Also, a relaxation technique may be usedin step f).

A sheet-like sensor for measuring stress distribution in accordance withthis invention may include a grid of members which change in resistancewhen subjected to strain, the members intersecting at internal nodes andintersecting at boundary nodes at the periphery of the grid defining aplurality of legs and means, such as an analyzer, connected only to theboundary nodes, for calculating any change in resistance in all of thelegs based solely on the measured resistance of the legs between theboundary nodes.

BRIEF DESCRIPTION OF THE DRAWINGS

Other objects, features and advantages will occur to those skilled inthe art from the following description of a preferred embodiment and theaccompanying drawings, in which:

FIG. 1 is a schematic view of a proposed grid showing the requirement ofelectrical interconnections between the analyzer and the internal nodesof the grid;

FIG. 2 is a schematic three dimensional view showing the prior artsheet-like sensor of U.S. Pat. No. 5,650,570;

FIG. 3 is a schematic view showing how, in the sensor of the subjectinvention, no electrical interconnections need exist between theanalyzer and the internal nodes of the grid;

FIG. 4 is a schematic view of a larger grid in accordance with thesubject invention;

FIGS. 5A-5B are schematic views showing other grid configurations inaccordance with the subject invention;

FIG. 6 is a flow chart depicting the primary steps associated with themethod of calculating strains in accordance with the subject invention;

FIG. 7 is a schematic cross-sectional view showing a sensor inaccordance with the subject invention disposed on an aircraft wing formeasuring stresses experienced thereby;

FIG. 8 is an explanatory circuit diagram showing a simple network ofresistors in series;

FIG. 9 is an explanatory circuit diagram showing resistors in paralleland in series;

FIG. 10 is a circuit diagram showing a grid with complex impedances;

FIG. 11 is a circuit diagram showing a twelve node resistive grid;

FIG. 12 is a view of several graphs showing the number of iterationsrequired of the algorithm of this invention in order to obtainconvergence when a relaxation technique is employed;

FIG. 13 is a view similar to FIG. 12 except no relaxation technique isemployed;

FIG. 14 is another set of graphs showing convergence in accordance withthe subject invention with the relaxation technique when a resistiveelement is removed;

FIG. 15 is a view similar to FIG. 14 except that the relaxationtechnique is not employed;

FIG. 16 is a set of graphs showing convergence for the real parts of thecomplex impedance grid shown in FIG. 10;

FIG. 17 is a set of graphs showing convergence of the imaginary parts ofthe complex impedance grid of FIG. 10;

FIG. 18 is a view of another resistive network which can be analyzed inaccordance with the subject invention;

FIG. 19 is a set of graphs showing convergence of the real parts of thegrid shown in FIG. 18;

FIG. 20 is a set of graphs showing the convergence of the imaginaryparts of the grid of FIG. 18;

FIG. 21 is a view of still another grid which can be analyzed inaccordance with the subject invention;

FIG. 22 is a set of graphs showing convergence of the real parts of thegrid of FIG. 21;

FIG. 23 is a set of graphs showing convergence of the imaginary parts ofthe grid of FIG. 21;

FIG. 24 is a view of another grid which can be fully analyzed inaccordance with the subject invention;

FIG. 25 is a set of graphs showing convergence of the real parts of thecomplex impedance shown in FIG. 24;

FIG. 26 is a set of graphs showing convergence of the imaginary parts ofthe complex impedances of the grid of FIG. 24;

FIG. 27 is a schematic view of a portion of a complete sheet-like sensorin accordance with the subject invention;

FIG. 28 is a schematic view of a portion of still another sheet-likesensor in accordance with the subject invention employing apiezoelectric actuator;

FIGS. 29-31 are three dimensional graphs showing various surface strainshapes;

FIG. 32 is a schematic view showing an embodiment of the subjectinvention wherein the sheet-like sensor is employed as a shear stresssensor on the hull of a ship;

FIG. 33 is a depiction of the deformation of the sensor of FIG. 32 undershear stress;

FIG. 34 is a stress-strain curve at different temperatures relative totransformation for various materials useful as the grid members inaccordance with the subject invention;

FIG. 35 is a graph showing transformation versus temperature for aspecimen under constant load;

FIG. 36 is a graph showing the pseudoelasticity of shape memory alloysused in accordance with the subject invention; and

FIGS. 37-38 are graphs showing wire resistance as a function of strainfor various pseudoelastic shape memory alloys useful in accordance withthe subject invention.

DISCLOSURE OF THE PREFERRED EMBODIMENTS

Aside from the preferred embodiment or embodiments disclosed below, thisinvention is capable of other embodiments and of being practiced orbeing carried out in various ways. Thus, it is to be understood that theinvention is not limited in its application to the details ofconstruction and the arrangements of components or steps set forth inthe following description or illustrated in the drawings.

Proposed sensor 10, FIG. 1 includes grid 12 of members 14 which changein resistance when subjected to strain. Node 5 is an “internal” node,nodes 1-4 and 6-9 are “boundary” nodes. By connecting wires or otherelectrical interconnects to each node (see wire 16 connected betweennode 2 and analyzer 18, and wire 17 connected between node 5 andanalyzer 18), analyzer 18 can be configured to calculate any change inresistance due to strain experienced by each leg of the grid andultimately the strain experienced by each leg.

As delineated in the Background section above, one problem with thisarrangement is the need for wires or electrical interconnects connectedto all of the nodes. For N×M nodes (in FIG. 1, 3×3 nodes) there must atleast N×M wires. In a practical system, N and M may each be 100 or moreresulting in at least 10,000 wires or electrical interconnections. Whensuch a system is formed as sheet-like sensor disposed on or integralwith a structure, such as a large aircraft wing, the large number ofelectrical interconnections becomes unwieldy as does the computationsrequired to be carried out by analyzer 18 to measure the change inresistance of all of the legs between all of the nodes.

Note, however, that for a system with 10,000 nodes, there are only2M+2N−4 boundary nodes or 396 boundary nodes. If any change inresistance of all the legs between the 10,000 nodes could be detectedvia wires or electrical interconnections connected to only the 396boundary nodes, there would be 9,604 less wires or electricalinterconnections greatly reducing the complexity and cost of the system.

Disclosed in U.S. Pat. No. 5,650,570 is sheet-like sensor 20, FIG. 2with parallel amorphous iron-based alloy members 22 woven in glass clothlayer 24 and similar parallel members 26 woven in glass cloth layer 28but running perpendicular to members 22. Insulating sheet 30 separatesand electrically isolates layers 24 and 28 and the respective woven-inmembers. Synthetic rubber sheets 32 and 34 cover layers 24 and 28,respectively.

Note that no nodes are formed. Thus, when an electrical connection ismade between each end of each member 22 and between each end of members26 to separate analyzers (see FIG. 25 of the '570 patent), any change inresistance along the length of a given member due to stress can bedetected but not the specific location of the stress in all cases, forexample, if only one member of layer 28 experiences stress. The same istrue if one member fails at some point along its length: the system cannot then identify the specific location of the failure. Moreover, thesystem of the '570 patent cannot accurately predict the full stressdistribution since it only provides an estimate of where a force orpressure is applied. Also, the need for the insulating sheet between thetwo glass cloth layers results in a thicker and more unwieldy sheet-likesensor.

In the subject invention, in contrast, stress locations can be preciselydetermined and measured by only attaching wires, leads, or otherelectrical interconnections to the boundary nodes of the sheet-likesensor. As shown in FIG. 3, simplified sheet-like sensor 40 includes agrid of members which change in resistance when subject to strain.Unlike the system disclosed in the '570 patent, the members intersect atinternal node 5 and intersect at boundary nodes 1-4 and 6-9 at theperiphery of the grid and thus define legs having a variable resistanceas a function of strain. Thus, R_(ij) is the resistance of a leg betweennodes i and j. The members or legs interconnecting the nodes aretypically copper wires or wires made of a pseudoelastic shape memorymaterial such as Niton. The members may also be traces etched,sputtered, laser machined, or otherwise formed on a substrate.

Unlike the system shown in FIG. 1, no electrical connections need bemade to internal node 5 in order to determine the change in resistanceof all of the legs. Thus, in a typical system, with, for example, 10,000total nodes, there are only 396 boundary nodes and thus at least 9,604fewer connections than the system of FIG. 1. The result is a lesscomplex, more reliable, and easier to deploy sensor sheet.

Analyzer 44 is configured to calculate any change in resistance in allof the legs based solely on the measured resistances of the legs betweenthe boundary nodes. For example, in system 40′, FIG. 4, electricalinterconnections need not be connected to internal nodes 7-9, 12-14, or17-19 in order to accurately calculate any change in resistance of anyof the legs including the legs between the internal nodes or between aboundary node and an internal node.

The subject invention is not limited to polygonal, rectangular, orsquare configurations, however. In system 40″, FIG. 5A, no electricalinterconnections are required for internal nodes 5, 9, 8, 12, 7 and 11in order to calculate any change in resistance of all of the legs of thegrid shown. In FIG. 5B, another circular configuration is shown with asingle internal node 9.

Returning again to FIG. 3 for simplicity and to FIG. 6, means such asanalyzer 44, FIG. 3, or any computation device or set of devices orequivalent, is electrically connected only to boundary nodes 1-4 and 6-9as shown and is configured, programmed, or adapted to first measure theresistance of the legs between the boundary nodes represented by R₁₂,R₂₃, R₃₆, R₆₉, R₈₉, R₇₈, R₄₇, and R₁₄, step 50, FIG. 6. Estimates arethen made for the resistances of all of the legs including the legsrepresented as R₂₅, R₄₅, R₅₆, and R₅₈ in FIG. 3, step 52, FIG. 6.Preferably, the estimates are made in such a way that they are as closeas possible to the actual leg resistances to advance convergence. Onemethod for making initial estimates for R₂₅, R₄₅, R₅₆, and R₅₈ and theother leg resistances includes setting them all to the mean of themeasured leg resistances between the boundary nodes.

Next, by knowing a) the resistive network or grid configuration andlayout, b) the measured resistances of the legs between the boundarynodes, and c) the estimated resistances of all of the legs, theresistance of all of the legs is calculated, step 54, FIG. 6, using i)the fact that at each node the net current flow is zero and that ii) thevoltage drop is a known function of resistance and current for aparticular leg material.

Then, the calculated resistances of the legs between the boundary nodesis compared, step 56, with the measured resistance of the legs betweenthe boundary nodes. Based on this comparison, at step 60, a new estimateis made for the resistances of all of the legs. If convergence has notoccurred at step 58, steps 54, 56, 58, and 60 are again carried outuntil convergence occurs whereupon the measured resistances of the legsbetween the boundary nodes (R₁₂, R₂₃, R₃₆, R₆₉, R₈₉, R₇₈, R₄₇, and R₁₄,FIG. 3) at step 50, FIG. 6 converge to the calculated resistances ofthese same legs. When this occurs, the calculated resistances of thelegs connected to any internal node (legs R₄₅, R₂₅, R₅₆, and R₅₈, FIG.3) or between internal nodes (see FIGS. 4-5) are accurately determined.

Once all the resistances are known and, more specifically, when there isa change in resistance of one or more of the legs due to strain, theresulting strain can be easily calculated step 62, FIG. 6. Also, if anyleg resistances are determined to be extremely high or infinity, afailed leg condition can be identified.

In a typical system, the sheet-like sensor of FIGS. 3-5 as representedby sensing grid 70, FIG. 7 is encapsulated in a flexible encapsulationmaterial such as Kapton layers 72 and 74 and laid on a structure whosestress distribution is to be measured, for example, aircraft wing 76.Indeed, the circuitry of analyzer 44, FIG. 3 may be integrated withKapton layer 72 and/or 74. And, the complete sensor can be integratedwith the structure itself. For example, if wing 76 is made of plies ofcomposite material, grid 70 can be integrated as one of the plies.

Examples of the Analysis Methods

This section provides the theory of how the internal voltages andcurrents in a grid with known leg resistances can be determined. Makinguse of two simple examples best does this. Consider the simple networkof resistors in series of FIG. 8.

Using standard electrical network theory, the following equations hold.

The voltage drop over Resistance R₁₂ is:

ΔV ₂₁ =V ₂ −V ₁ =I ₁ R ₁,  (1)

the voltage drop over Resistance R₂₃ is:

V ₃ −V ₂ =I ₂ R ₂, and  (2)

the voltage drop over Resistance R₃₄ is:

V ₄ −V ₃ =I ₃ R ₃.  (3)

At the nodes, the net current flow must be zero. Thus, at node 2

I ₁ −I ₂=0, and  (4)

at node 3,

I ₂ −I ₃=0  (5)

The unknowns I₁, I₃, I₄, V₂, V₃ can be determined from equations (1)through (5) when a voltage is applied across nodes 1 and 4 and when V₁and V₄ are known. Equations (1) through (5) can be placed into matrixform, which yields: $\begin{matrix}{{\begin{bmatrix}{- 1} & 1 & 0 & 0 \\0 & {- 1} & 1 & 0 \\0 & 0 & {- 1} & 1 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0\end{bmatrix}\begin{Bmatrix}V_{1} \\V_{2} \\V_{3} \\V_{4}\end{Bmatrix}} = {\begin{bmatrix}R_{1} & 0 & 0 \\0 & R_{2} & 0 \\0 & 0 & R_{3} \\1 & {- 1} & 0 \\0 & 1 & {- 1}\end{bmatrix}\begin{Bmatrix}I_{1} \\I_{3} \\I_{4}\end{Bmatrix}}} & (6)\end{matrix}$

In order to solve this set of equations for the unknowns, the equationsare re-organized as shown in Equation (7): $\begin{matrix}{{\begin{bmatrix}1 & 0 & {- 1} & 0 \\{- 1} & 1 & 0 & 0 \\0 & {- 1} & 0 & 1 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0\end{bmatrix}\begin{Bmatrix}V_{2} \\V_{3} \\V_{1} \\V_{4}\end{Bmatrix}} = {\begin{bmatrix}R_{1} & 0 & 0 \\0 & R_{3} & 0 \\0 & 0 & R_{4} \\1 & {- 1} & 0 \\0 & 1 & {- 1}\end{bmatrix}\begin{Bmatrix}I_{1} \\I_{3} \\I_{4}\end{Bmatrix}}} & (7)\end{matrix}$

and then the unknowns can be calculated from: $\begin{matrix}{{\left\lbrack {\begin{matrix}\begin{bmatrix}1 & 0 \\{- 1} & 1 \\0 & {- 1}\end{bmatrix} \\\begin{bmatrix}0 & 0 \\0 & 0\end{bmatrix}\end{matrix}\begin{matrix}\begin{bmatrix}{- 1} & 0 \\0 & 0 \\0 & 1\end{bmatrix} \\\begin{bmatrix}0 & 0 \\0 & 0\end{bmatrix}\end{matrix}} \right\rbrack \begin{Bmatrix}V_{2} \\V_{3} \\V_{1} \\V_{4}\end{Bmatrix}} = {{\begin{bmatrix}\begin{bmatrix}R_{1} & 0 & 0 \\0 & R_{2} & 0 \\0 & 0 & R_{3}\end{bmatrix} \\\begin{bmatrix}1 & {- 1} & 0 \\0 & 1 & {- 1}\end{bmatrix}\end{bmatrix}{\begin{Bmatrix}I_{1} \\I_{3} \\I_{4}\end{Bmatrix}\quad \begin{bmatrix}A_{11} & A_{12} \\A_{21} & A_{22}\end{bmatrix}}\begin{Bmatrix}V_{u} \\V_{k}\end{Bmatrix}} = {\begin{bmatrix}B_{1} \\B_{2}\end{bmatrix}\left\{ I_{u} \right\}}}} & (8)\end{matrix}$

re-organized to

A ₁₁ V _(u) −B ₁ I _(u) =−A ₁₂ V _(k)

A ₂₁ V _(u) −B ₂ I _(u) =−A ₂₂ V _(k)  (9)

$\begin{matrix}{{{{or}\quad\begin{bmatrix}A_{11} & {- B_{1}} \\A_{21} & {- B_{2}}\end{bmatrix}}\begin{Bmatrix}V_{u} \\I_{u}\end{Bmatrix}} = {{- \begin{bmatrix}A_{12} \\A_{22}\end{bmatrix}}\left\{ V_{k} \right\}}} & (10) \\{{{with}\quad \begin{Bmatrix}V_{u} \\I_{u}\end{Bmatrix}} = {{- \quad {\begin{bmatrix}A_{11} & {- B_{1}} \\A_{21} & {- B_{2}}\end{bmatrix}^{- 1}\begin{bmatrix}A_{12} \\A_{22}\end{bmatrix}}}\left\{ V_{k} \right\}}} & (11)\end{matrix}$

Equation (11) provides the solution for the unknowns in the grid.

Another example is a circuit with resistors in parallel and in series asshown in FIG. 9.

A set of equations to solve for the unknowns (I₁, I₂, I₃, I₄, V₂, V₃)can be formed as before.

The voltage drop over R₁ (over network nodes 1 and 2) is:

V ₂ −V ₁ =I ₁ R ₁,  (12)

the voltage drop over R₂ (over network nodes 2 and 3) is:

V ₃ −V ₂ =I ₂ R ₂,  (13)

the voltage drop over R₃ (over network nodes 2 and 3) is:

V ₃ −V ₂ =I ₃ R ₃, and  (14)

the voltage drop over R4 (over network nodes 2 and 3) is:

V ₄ −V ₃ =I ₄ R ₄.  (15)

At the nodes, the net current flow must be zero. Thus, at node 2

I ₁ −I ₂ −I ₃=0, and  (16)

at node 3

I ₂ +I ₃ −I ₄=0.  (17)

In matrix form: $\begin{matrix}{{\begin{bmatrix}{- 1} & 1 & 0 & 0 \\0 & {- 1} & 1 & 0 \\0 & {- 1} & 1 & 0 \\0 & 0 & {- 1} & 1 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0\end{bmatrix}\begin{Bmatrix}V_{1} \\V_{2} \\V_{3} \\V_{4}\end{Bmatrix}} = {\begin{bmatrix}R_{1} & 0 & 0 & 0 \\0 & R_{2} & 0 & 0 \\0 & 0 & R_{3} & 0 \\0 & 0 & 0 & R_{4} \\1 & {- 1} & {- 1} & 0 \\0 & 1 & 1 & {- 1}\end{bmatrix}\begin{Bmatrix}i_{1} \\i_{2} \\i_{3} \\i_{4}\end{Bmatrix}}} & (18)\end{matrix}$

This matrix can be re-organized and solved following the steps shown inEquations (8) through (11).

Thus, one innovation of this invention is a method for determining theleg resistances of a grid network of resistances. The internal currentsand voltages can be determined by applying known voltages to theexternal nodes of the network if the leg resistances are known using aniterative algorithm that converges to the leg resistances, without anyprior knowledge of the resistances.

The network of FIG. 3 will be used as an example to describe thepreferred method. In this 12 leg resistive network, nodes 1-4 and 6-9are boundary nodes. All of the leg resistances are determined by onlymeasuring the resistances between the boundary nodes. The algorithmpreferably places these resistances in a column vector {R_(Meas)} thus:$\begin{matrix}{\left\{ \Re_{Meas} \right\} = \begin{matrix}\quad & \begin{matrix}\left\lbrack R_{12} \right. & R_{13} & R_{14} & R_{16} & R_{17} & R_{18} & R_{19} \\\quad & R_{23} & R_{24} & R_{26} & R_{27} & R_{28} & R_{29} \\\quad & \quad & R_{34} & R_{36} & R_{37} & R_{38} & R_{39} \\\quad & \quad & \quad & R_{46} & R_{47} & R_{48} & R_{49} \\\quad & \quad & \quad & \quad & R_{67} & R_{68} & R_{69} \\\quad & \quad & \quad & \quad & \quad & R_{78} & R_{79} \\\quad & \quad & \quad & \quad & \quad & \quad & \left. R_{89} \right\rbrack\end{matrix}\end{matrix}} & (19)\end{matrix}$

Note that R_(ij) is the measured resistance or impedance betweenboundary nodes i and j.

The first step in the iterative algorithm is to calculate what theresistances between the boundary nodes will be for initial estimates ofthe leg resistances (R_(Estimate) ^(k)). Here k is the index of theiterative loop. The closer the initial estimates are to the actual legresistances, the quicker the algorithm will converge. Intelligentmethods for determining good initial estimates are discussed later.

The calculated resistances are obtained by using the method outlined inthe previous section. Once the input currents are known, the resistancesbetween boundary nodes can be calculated and stored in a column vector{R_(Estimate) ^(k)}.

The second step in the iterative algorithm is to determine how thiscolumn of resistances will change when a leg resistance is perturbed.Let {R_(n) ^(k)} be the column vector between nominal and when legnumber n's resistance has been changed by a small delta from the initialestimate, that is let: $\begin{matrix}{\left\{ \Re_{n}^{k} \right\} = {\left\{ {\Re \begin{pmatrix}\begin{matrix}\begin{matrix}R_{1}^{k} \\M\end{matrix} \\{R_{n}^{k}\left( {1 + \delta} \right)}\end{matrix} \\M\end{pmatrix}} \right\} - \left\{ \Re_{Estimate}^{k} \right\}}} & (20)\end{matrix}$

A matrix [R_(Perturbed) ^(k)] is constructed of these column vectorswhere the columns are obtained by varying sequentially the legresistances.

[R _(Perturbed) ^(k) ]=[{R ₁ ^(k) }{R ₂ ^(k) }L{R _(Nelem) ^(k)}]  (21)

where Nelem is the number of legs (resistive elements) in the grid. Animproved estimate of the leg resistances are obtained using thefollowing equation: $\begin{matrix}{R_{Estimate}^{k + 1} = {R_{Estimate}^{k} + {F_{Relax}\delta \quad {R_{Estimate}^{k} \cdot {{\left\lbrack {\left\lbrack \Re_{Perturbed}^{k} \right\rbrack^{T}\left\lbrack \Re_{Perturbed}^{k} \right\rbrack} \right\rbrack^{- 1}\left\lbrack \Re_{Perturbed}^{k} \right\rbrack}^{T}\left\lbrack {\Re_{Estimate}^{k} - \Re_{Meas}} \right\rbrack}}}}} & (22)\end{matrix}$

Where F_(Relax) is a relaxation factor determined by standard relaxationmethods.

The algorithm can also be used to identify failures in electroniccircuits. Since the algorithm will sense the impedance of grid elements,the algorithm can detect when a leg is short circuited, the resistanceis zero, or near-zero. The algorithm can also detect if a connection hasbeen broken. For broken connections, the leg impedance goes to infinity,which can be detected by the algorithm.

The algorithm can also be used to identify complex impedances in anetwork. For example, by using complex variables, the approach can beused to identify complex impedances of the form:

Z=Z _(R) +jZ ₁  (23)

where j={square root over (−1)}, Z_(R) is the real component and Z₁ isthe imaginary component of the impedance. In this process, an impedancemeter will measure the complex impedance across the external nodes andthe same procedure outlined in equations (19) through (22) will yield ameasure of the grid impedances.

By measuring the impedance of the network at different frequencies, themethod can be used to identify capacitances and inductors in anelectrical network. Thus each leg can have a combination of a pureresistance, a pure capacitor and/or a pure inductor as shown in FIG. 10.When a leg impedance is measured at three different frequencies ω₁, ω₂,and ω₃, three equations can be written for the unknowns in the legimpedance (R, C and L): $\begin{matrix}\begin{matrix}\begin{matrix}{{{j\quad \omega_{1}L} + R - {j\frac{C}{\omega_{1}}}} = {Z_{1} = {Z\left( \omega_{1} \right)}}} \\{{{j\quad \omega_{2}L} + R - {j\frac{C}{\omega_{2}}}} = {Z_{2} = {Z\left( \omega_{2} \right)}}}\end{matrix} \\{{{j\quad \omega_{3}L} + R - {j\frac{C}{\omega_{3}}}} = {Z_{3} = {Z\left( \omega_{3} \right)}}}\end{matrix} & (24)\end{matrix}$

where Z₁, Z₂, and Z₃ are the leg impedances obtained at the threedifferent frequencies. From these three equations the equivalentresistance, capacitance and inductance of any leg can be uniquelydetermined. When more test frequencies are used, minimization techniquescan be used to estimate the values of leg elements.

The method or algorithm can use all the nodes in the grid or only theboundary nodes. The question is if all the leg impedances can bedetermined if resistances are only measured between the boundary nodes.

For a regular grid of N×M nodes, the number of boundary nodes is

2M+2N−4, and  (25)

the number of elements in a regular grid (See FIGS. 3 and 10) is

M(N−1)+N(M−1)=2MN−M−N,  (26)

and the number of leg impedances that can be determined is equal to thenumber of unique measurements that can be made between the externalnodes is:

[2N+2M−4][2N+2M−4−1]/2=2N ²+2M ²+4NM−9N−9M+10  (27)

Since

(2N ²+2M ²+4NM−9N−9M+10)−(2MN−M−N)=2(N ² +M ² +MN)−8(N+M)+10>0,  (28)

the conclusion is that additional elements can be added to the regulargrid.

A good choice for an initial guess is the mean of the measuredresistances between the boundary nodes, that is:

R_(n) ^(k=1)=mean{R_(Meas)}for n=1,Nelem  (29)

This choice leads to convergence without any relaxation in most casesstudied.

The grid of FIG. 11 was constructed using carbon resistors. Resistancemeasurements between the boundary nodes (see equation (19)) are reportedin Table I. The algorithm was coded in Matlab, and converged rapidly tothe actual leg resistances (Table II). Convergences of two cases areshown in FIGS. 12 and 13. FIG. 12 shows how the algorithm converges whenrelaxation is used and FIG. 13 shows the convergence when no relaxationis used in the algorithm. Note that although not shown, convergence isachieved within one or two steps when the initial guesses of the legresistances are near the actual values.

TABLE I Measured Resistances between Boundary Nodes.

TABLE II The resistances as determined by the Iterative Algorithm.Resistance from Resistance Algorithm (Ω) (Ω) 219 219 428 428 243 243 699700 388 389 219 219  36  35 682 683  91  91  30  30 239 238  99  98

In another experiment, resistance R₇ was removed in the grid of FIG. 11to show that the method can be used to determine failure of components.Resistance measurements between the boundary nodes (see equation (19))are reported in Table III. The algorithm again converged rapidly to theactual leg resistances (Table IV). Convergences of two cases are shownin FIGS. 14 and 15. FIG. 14 shows how the algorithm converges whenrelaxation is used and FIG. 15 shows the convergence when no relaxationis used in the algorithm.

All unknown resistances with one removed

TABLE III Measured Resistances between Boundary Nodes.

TABLE IV The resistances as determined by the Iterative Algorithm forthe case where Resistor 7 is removed (Open Circuit between Nodes 5 and6). Resistance from Resistance Algorithm (Ω) (Ω) 219 216 428 426 243 241699 709 388 389 219 219 Open Open 682 677  91  91  30  30 239 239  99 99

A mathematical model was also constructed of the circuit shown in FIG.10. The leg impedances were randomly selected as shown in Table V below:

TABLE V  R1 = (62.6 + 13.1j) Ω  R2 = (103.2 + 61.8j) Ω  R3 = (14.3 +−128.0j) Ω  R4 = (217.4 + −52.9j) Ω  R5 = (128.2 + 124.7j) Ω  R6 =(93.2 + 79.6j) Ω  R7 = (113.4 + −17.1j) Ω  R8 = (73.5 + 29.8j) Ω  R9 =(95.2 + 129.5j) Ω R10 = (49.7 + −194.0j) Ω R11 = (127.6 + 45.3j) Ω R12 =(167.3 + −197.9j) Ω

The convergence of the algorithm, starting with the mean of thesimulated measurements between the boundary nodes are shown in FIGS. 16and 17. FIG. 16 is the convergence of the real parts and FIG. 17 theconvergence of the imaginary parts of the leg impedances. Note thatalthough not shown, convergence is achieved within one or two steps whenthe initial guesses of the leg impedances are near the actual values.

In another example, a mathematical model was constructed of the circuitshown in FIG. 18. FIGS. 19 and 20 show that the algorithm converges tothe correct impedances.

In another example, a mathematical model was constructed of the circuitshown in FIG. 21. FIGS. 22 and 23 show that the algorithm converges tothe correct impedances.

Another mathematical model was constructed of the 16 node circuit shownin FIG. 24. The leg impedances were randomly selected as:

TABLE VI  R1 = (1.3 + 199.9j) Ω  R2 = (126.4 + −162.3j) Ω  R3 = (163.7 +−190.0j) Ω  R4 = (177.5 + −164.9j) Ω  R5 = (140.3 + −146.9j) Ω  R6 =(55.3 + 181.0j) Ω  R7 = (31.8 + −160.0j) Ω  R8 = (143.3 + 51.5j) Ω  R9 =(208.1 + −101.6j) Ω R10 = (179.5 + −122.9j) Ω R11 = (204.7 + 93.7j) ΩR12 = (68.2 + 21.6j) Ω R13 = (59.1 + 194.2j) Ω R14 = (118.0 + −74.9j) ΩR15 = (35.9 + 90.0j) Ω R16 = (46.4 + 195.1j) Ω R17 = (47.7 + 35.9j) ΩR18 = (143.4 + 167.3j) Ω R19 = (11.6 + 109.8j) Ω R20 = (50.4 + −53.0j) ΩR21 = (146.8 + 99.2j) Ω R22 = (68.4 + −148.4j) Ω R23 = (67.5 + 200.7j) ΩR24 = (158.5 + −133.7j) Ω

The convergence of the algorithm, starting with the mean of thesimulated measurements between the boundary nodes are shown in FIGS. 25and 26. FIG. 25 shows the convergence of the real parts and FIG. 26shows the convergence of the imaginary parts of the leg impedances. Notethat although not shown, convergence is achieved within one or two stepswhen the initial guesses of the leg impedances are near the actualvalues.

In addition, although leg resistances and impedances are the typicalcharacteristics analyzed by the method of this invention, analysis ofother characteristics is possible.

Examples of Sensor Technologies

Strains induced in a structure are often complex and, depending on theloads, the strains can vary significantly spatially. This inventionprovides a grid of resistive elements that are sensitive to changes instrain. For example, constructing a grid of copper wires, where theresistance of the copper wires changes when they are strained, can formsuch a grid. An alternative strain sensitive material is pseudoelasticshape memory alloy wires. Using the methods described above, theresistance of each leg in the sensor grid is determined and the sensorgrid can be used to measure, in detail, the complex strain in astructure. The sensors can be large to measure global structural strainsor small to obtain a detail measurement of a stress concentration.

The sensing grids can also be used to determine spatial variation ofloads and to determine the point of application of a load, for example,where a finger or stylus presses down on an input device constructedusing the strain sensitive grid.

In one aspect of this invention is a grid of strain sensitive elementsare used and the algorithm determines the resistance in each leg of thegrid while only having electrical access to the external or boundarynodes of the grid.

The effective algorithms described above are used to determine theresistances in a sensor-grid make strain-sensing grids attractive formany applications. The grid elements in the sensing grid are connectedat the internal nodes. FIG. 11 serves as an example. In this resistivegrid, the algorithm only requires the resistances between the boundarynodes to be measured. For example: R1-2 (the resistance between Node 1and 2), R1-3, R1-4, R1-6, R1-7, R1-8, R1-9, R2-3, R2-3, etc.

The algorithm converged to the actual resistance values within 8iterations. It should be noted that algorithm converges much faster (oneor two iterations) when the initial guess for the grid resistances arenear the actual values. In this example the initial guess was randomlyset to be between 0 and 500 Ω. Given that in strain sensors theresistance change is proportional to strain, the conclusion is that thealgorithm can thus measure the change in strain anywhere in thesensor-grid.

And, when Resistor 7 was removed, simulating a destructive failure atthis location, the algorithm, using only measurements made throughelectrically accessing the boundary nodes (1, 2, 3, 4, 6, 7, 8 and 9),rapidly converges to the correct resistances. By verifying that thealgorithm can detect a broken connection, it is demonstrated that theproposed solution is not only robust, but it can also be used forstructural health monitoring.

The preferred sensor grid of FIG. 27 is encapsulated in an appropriateencapsulation material. Kapton is one choice, but other materials thatare used in standard foil and ceramic strain gauges can also be used.Encapsulation provides a robust solution that will allow the sensor tobe easily bonded to structural surfaces or to be integrated as acomposite layer.

Structural strain will be inferred from the measurement of the change ingrid-leg resistances. The “leg” resistances are determined by thealgorithm disclosed above that only requires access to the boundarynodes of the grid.

The use of a strain-sensing grid of strain-sensitive thin shape memoryalloy wires 100 connected on their boundary ends to bus 102 allow thesensor-grid to measure large strains. Depending on the fineness of thegrid, strain can be monitored in far more detail than possible withpoint sensors. Ultra-thin copper, pseudoelastic shape memory alloywires, and standard foil or ceramic gauges can be used as the sensorelements (legs).

The low-weight, robust and thin encapsulated sensor grid can be bondedto any structural surface, or it can be integrated into the structureitself as one of the layers in the composite. Logic can be added to thesensor-grid to determine structural health, structural shape, straindetail at stress concentrations, and structural dynamic response by theuse of circuitry disposed on flex circuit 104. The sensor grids can alsobe used to improve the accuracy and sensitivity of loadcells, pressuresensors and accelerometers.

When piezoelectric actuator 106, FIG. 28 is added to the sensor-grid,the sensor-grid can also be used to sense structural vibration foractive vibration and noise control and structural health monitoring.Using a piezoelectric actuator to create vibrations in the structure,the sensor-grid can be used to detect the structural response.

The proposed technology is markedly different from the “point-sensor”approach and warrants some basic understanding of the problem. In orderto make the case that the proposed technology is feasible, a simpleRayleigh-Ritz model of a 1 m span, 0.4 m chord and 20 mm thickcantilevered plate was constructed. This Rayleigh-Ritz model was used topredict the static deflection of the plate when subjected to out-boardleading and trailing edge vertical loads. The vertical loads were sizedto yield a maximum deflection equal to one percent (1%) of the spandimension, namely 10 mm. FIGS. 29-31 show the surface strains predictedby this model for these loads. The surface strains are measured bystrain gauges attached to the upper or lower surface of the plate areapproximately 340 microStrain for this moderate deflection.State-of-the-art data acquisition systems can reliably measure strainsin the 10-20 microStrain range.

Embedding the sensor-grid in a soft polymer also allows the subjectinvention to be used as a fluid flow shear sensor. When the sensor isattached to the surface of an underwater vehicle, it can measure shearstresses induced by fluid flow.

A shear-sensor design is shown in FIG. 32. In this design both ends ofthe sensor are fixed. The strain-sensitive wires will be pre-stressedbetween the two “fixed” boundaries while the sides will beunconstrained. The strain-sensitive wires will be pre-stressed toaccommodate the compressive stresses that will be seen by the wire. Thefinite element model of FIG. 33 predicts a sensitivity of 0.1microStrain/Pa.

FIG. 34 illustrates typical stress-strain curves for shape memory alloymaterials in a test set-up (e.g. a wire made of a shape memory alloy).For illustration purposes, curves for pseudoelastic and martensitephases are included (where only one of the pseudoelastic and martensitephases is present for a given material). For a material with anaustenite phase present at T₁ and a martensite phase at T₂, the materialcan be strained by approximately 3%-8% of its length under low appliedstresses. If the temperature of the alloy material is raised above itstransition temperature, the material changes to its austenite phase andrecovers to its original, underformed shape. This transformation isshown in FIG. 35, in which A_(s) and A_(f) represent start and finishpoints of the austenite phase, respectively; and M_(s) and M_(f)represent start and finish points of the martensite phase, respectively.

According to FIG. 35, for a shape memory alloy material at equilibriumin the austenite phase, when a dynamic stress is applied, the materialis converted from austenite to martensite as the temperature drops belowM_(s), while the length of the SMA wire increases until the temperaturereaches the M_(f) temperature. As indicated, the transition from theaustenite to the martensite phases is reversible. By heating the SMAmaterial, its length decreases between A_(s) and A_(f) until thematerial recovers to its original pre-stressed length.

FIG. 36 illustrates the property of pseudoelasticity, which is presentin certain SMAs. For a pseudoelastic alloy material, the pseudoelasticphase is a type of martensite phase in which deformation can occur. Asseen in FIG. 36, the material can reversibly strain by up toapproximately 8%. As with non-pseudoelastic materials, the pseudoelasticalloy follows a different return path to the austenite phase, indicatingthat the material absorbs energy during the transformation.

Because pseudoelastic alloys and shape memory alloys exhibit measurablechanges of resistance when strained, such alloys are suitable for use instrain gauges/sensors. FIG. 37 depicts the resistance change of aNitinol wire in a test set-up. A pseudoelastic Nitinol wire 55 cm longand 1 mm in diameter was clamped at either end, and stresses wereapplied which produced the strain levels indicated on the graph. Achange of resistance was measured using conventional strain measurementtechniques, e.g. by subjecting the wire to a current. As indicated inFIG. 37, the Nitinol wire in pseudoelastic form reversibly elongated byapproximately 5% without permanent deformation of the wire. Such a wirecan be incorporated into the strain gauges of the present invention.

By contrast, FIG. 38 illustrates the resistance change of anon-pseudoelastic shape memory alloy material, in a test set-up similarto that described with reference to FIG. 37. A shape memory Nitinol wire30 cm long and 1 mm in diameter in the martensite state was tested. TheNitinol wire experienced a change of resistance upon stretching, with aresistance change somewhat less than the pseudoelastic alloy. Whenstrained to a similar threshold as the pseudoelastic wire of FIG. 37,approximately 1.5% of strain was plastic deformation, and could not berecovered without heating the wire. Thus, while non-pseudoelastic shapememory alloys can be used in strain gauges according to the presentinvention, they must be heated in order to recover any plasticdeformation if reuse is desired. Alternatively, such materials can bedesigned for single use applications such as cargo loading systems inwhich a load is tested to determine whether any load shifting isacceptable. Generally, pseudoelastic alloys are preferred for reuseapplications because they permit maximum strain recovery without plasticdeformation.

A strain gauge incorporating a pseudoelastic alloy material functions ina manner similar to conventional strain gauges, except that it iscapable not only of measuring small strains in an object, but alsomedium to large size strains because of the use of a pseudoelastic alloymaterial. Conventional strain gauges made of typical metals and metalalloys fail upon straining with approximately 0.1-1% elongation, whereasthe present invention is directed to strain gauges made of pseudoelasticmaterials capable of withstanding approximately 8% elongation withoutpermanent deformation.

This invention thus provides an improved sheet-like sensor for measuringstress distribution. The number of electrical connections required tofully analyze the strain experienced by a structural component isseriously reduced because no wires need be connected internally to thesensor to fully analyze the stress distribution. The sensor is capableof detecting the specific location of any strains experienced and iscapable of identifying the specific location of a failure. The sensor isable to measure strains of a higher magnitude and can fully predictstress distribution. The method of this invention can be used inconnection with the sheet-like sensors disclosed or their equivalentsor, indeed, to identify failures of electronic circuitry. A better, lesscumbersome, more accurate, and more useful sheet-like sensor is effectedby arranging members which change resistance as a function of strain asa grid forming legs between both internal and external nodes but onlyconnecting the resistance measurement means or analyzer to the boundarynodes and then determining all of the leg resistances based on themeasured resistance of the legs between the boundary nodes. In this way,there need be no electrical interconnections between the analyzer andthe internal nodes of the grid thus seriously reducing the number ofelectrical interconnections required. Moreover, the specific location ofany strains experienced by the sensor can be more accurately detected,the specific location of any failure can be identified, and full stressdistribution of a structural member or component underlying the sensorcan be predicted. In addition, by using pseudoelastic shape memory alloymaterial instead of ferromagnetic materials, strains of a highermagnitude can be measured.

Although specific features of the invention are shown in some drawingsand not in others, this is for convenience only as each feature may becombined with any or all of the other features in accordance with theinvention. The words “including”, “comprising”, “having”, and “with” asused herein are to be interpreted broadly and comprehensively and arenot limited to any physical interconnection. Moreover, any embodimentsdisclosed in the subject application are not to be taken as the onlypossible embodiments.

Other embodiments will occur to those skilled in the art and are withinthe following claims:

What is claimed is:
 1. A sheet-like sensor for measuring stressdistribution comprising: a grid of members which change in resistancewhen subjected to strain, the members intersecting at internal nodes andintersecting at boundary nodes at the periphery of the grid defining aplurality of legs; and an analyzer connected to the boundary nodes andconfigured to measure any change in resistance of the legs between theboundary nodes and, based on the measured resistances, to calculate anychange in resistance in all of the legs.
 2. A sheet-like sensor formeasuring stress distribution comprising: a grid of members which changein resistance when subjected to strain, the grid of members encapsulatedin an encapsulation material and intersecting at internal nodes andintersecting at boundary nodes at the periphery of the grid defining aplurality of legs; and an analyzer circuit integral with theencapsulation material electrically connected only to the boundary nodesand configured to calculate any change in resistance in all of the legsbased solely on the measured resistance of the legs between the boundarynode.
 3. A sheet-like sensor for measuring stress distributioncomprising: a grid of members which change in resistance when subjectedto strain, the members intersecting at internal nodes and intersectingat boundary nodes at the periphery of the grid defining a plurality oflegs; and an analyzer connected only to the boundary nodes.
 4. Thesensor of claim 3 in which the analyzer is configured to: a) measure theresistances of the legs between the boundary nodes, b) estimate theresistances of all of the legs, c) calculate the resistances of all ofthe legs based on the measured resistances of the legs between theboundary nodes and the estimated resistances of all of the legs, d)compare the calculated resistances of the legs between the boundarynodes with the measured resistances of the legs between the boundarynodes, e) based on the comparison, re-estimate the resistances of all ofthe legs; and f) iteratively repeat steps c)-e) until the measuredresistances of the legs between the boundary nodes converge to thecalculated resistances of the legs between the boundary nodes to thusaccurately determine the resistances of the legs between or connected tothe internal nodes.
 5. A sheet-like sensor for measuring stressdistribution comprising: a grid of members which change in resistancewhen subjected to strain, the members intersecting at internal nodes andintersecting at boundary nodes at the periphery of the grid defining aplurality of legs; and means connected to the boundary nodes, forcalculating any change in resistance in all of the legs based solely onthe measured resistance of the legs between the boundary nodes.
 6. Asheet-like sensor for measuring stress distribution comprising: a gridof members which change in resistance when subjected to strain, themembers intersecting at internal nodes and intersecting at boundarynodes at the periphery of the grid defining a plurality of legs; and ananalyzer electrically connected only to the boundary nodes an configuredto calculate any change in resistance in all of the legs based solely onthe measure resistance of the legs between the boundary nodes.
 7. Thesensor of claim 6 in which the members are wires or traces.
 8. Thesensor of claim 7 in which the wires are made of copper.
 9. The sensorof claim 7 in which the wires are made of pseudoelastic shape memoryalloy material.
 10. The sensor of claim 6 in which the grid of membersis encapsulated in an encapsulation material.
 11. The sensor of claim 10in which the encapsulation material is Kapton.
 12. The sensor of claim10 in which the analyzer is a circuit integral with the encapsulationmaterial.
 13. The sensor of claim 6 in which the grid is in the shape ofa polygon.
 14. The sensor of claim 13 in which polygon is a rectangle.15. The sensor of claim 14 in which the rectangle is a square.
 16. Thesensor of claim 6 in which the analyzer is configured to: a) measure theresistances of the legs between the boundary nodes, b) estimate theresistances of all of the legs, c) calculate the resistances of all ofthe legs based on the measured resistances of the legs between theboundary nodes and the estimated resistances of all of the legs, d)compare the calculated resistances of the legs between the boundarynodes with the measured resistances of the legs between the boundarynodes, e) based on the comparison, re-estimate the resistances of all ofthe legs, and f) iteratively repeat steps c)-e) until the measuredresistances of the legs between the boundary nodes converge to thecalculated resistances of the legs between the boundary nodes to thusaccurately determine the resistances of the legs between or connected tothe internal nodes.
 17. The sensor of claim 16 in which the analyzer isfurther configured to calculate the strain experience by each leg. 18.The sensor of claim 17 in which the analyzer is further configured toidentify any leg which has failed based on a very high determinedresistance.
 19. The sensor of claim 16 in which the estimate is based onthe step a).
 20. The sensor of claim 19 in which the estimate is set tothe mean of the measured resistances.
 21. The sensor of claim 16 inwhich relaxation is used in step f).
 22. A sensor system for measuringstress distribution comprising: a grid of members which change inresistance when subjected to strain, the members intersecting atinternal nodes and intersecting at boundary nodes at the periphery ofthe grid defining a plurality of legs; and an analyzer electricallyconnected to the boundary nodes and configured to: a) measure theresistances of the legs between the boundary nodes, b) estimate theresistances of all of the legs, c) calculate the resistances of all ofthe legs based on the measured resistance of the legs between theboundary nodes and the estimated resistances of all of the legs, d)compare the calculated resistances of the legs between the boundarynodes with the measured resistances of the legs between the boundarynodes, e) based on the comparison, re-estimate the resistances of all ofthe legs, and f) iteratively repeat steps c)-e) until the measuredresistances of the legs between the boundary nodes converge to thecalculated resistances of legs between the boundary nodes to thusaccurately determine the resistances of the legs between or connected tothe internal nodes.
 23. A sensor system comprising: a grid includinginternal nodes and boundary nodes at the periphery of the grid defininga plurality of legs; and an analyzer electrically connected to theboundary nodes and configured to: a) measure a characteristic of thelegs between the boundary nodes, b) estimate the same characteristic ofall of the legs, c) calculate the same characteristic of all of the legsbase on the measured characteristic of the legs between the boundarynodes and the estimated characteristic of all of the legs, d) comparethe calculated characteristic of the legs between the boundary nodeswith the measured characteristic of the legs between the bound nodes, e)based on the comparison, re-estimate the resistances of all of the legs;and f) iteratively repeat steps c)-e) until the measured characteristicof the legs between the boundary nodes converge to the calculatedcharacteristic of the legs between the boundary nodes.
 24. The sensorsystem of claim 23 in which the members change in resistance whensubjected to strain and the characteristic is resistance.
 25. A methodfor determining impedances in a grid of leg impedances, the legsintersecting at boundary nodes at the periphery of the grid andintersecting at internal nodes within the grid, the method comprising:a) measuring the resistances of the legs between the boundary nodes, b)estimating the resistances of all of the legs, c) calculating theresistances of all of the legs based on the measured resistances of thelegs between the boundary nodes and the estimated resistances of all ofthe legs, d) comparing the calculated resistances of the legs betweenthe boundary nodes with the measured resistances of the legs between theboundary nodes, e) based on the comparison, re-estimating theresistances of all of the legs, and f) iteratively repeating steps c)-e)until the measured resistances of the legs between the boundary nodesconverge to the calculated resistances of the legs between the boundarynodes to thus accurately determine the resistance of the legs between orconnected to the internal nodes.
 26. The method of claim 25 furtherincluding the step of calculating the strain experience by each leg. 27.The method of claim 25 further including identifying any leg which hasfailed based on a very high determined resistance.
 28. The method ofclaim 25 in which the estimate of step b) is based on step a).
 29. Themethod of claim 25 in which the estimate is set to the mean of themeasured resistances.
 30. The method of claim 25 in which relaxation isused in step f).
 31. A sheet-like sensor for measuring stressdistribution comprising: a grid of wires which change in resistance whensubjected to strain, the wires intersecting at internal nodes andintersecting at boundary nodes at the periphery of the grid defining aplurality of legs, the wires encapsulated in an encapsulation material;and an analyzer connected to the boundary nodes and configured to: a)measure the resistances of the legs between the boundary nodes, b)estimate the resistances of all of the legs, c) calculate theresistances of all of the legs based on the measured resistances of thelegs between the boundary nodes and the estimated resistances of all ofthe legs, d) compare the calculated resistances of the legs between theboundary nodes with the measured resistances of the legs between theboundary nodes, e) based on the comparison, re-estimate the resistancesof all of the legs; and f) iteratively repeat steps c)-e) until themeasured resistances of the legs between the boundary nodes converge tothe calculated resistances of the legs between the boundary nodes tothus accurately determine the resistances of the legs between orconnected to the internal nodes.
 32. A method of measuring stressdistribution in a grid of wires which change in resistance whensubjected to strain, the wires intersecting at internal nodes andintersecting at boundary nodes at the periphery of the grid defining aplurality of legs, the method comprising: a) measuring the resistancesof the legs between the boundary nodes, b) estimating the resistances ofall of the legs, c) calculating the resistances of all of the legs basedon the measured resistances of the legs between the boundary nodes andthe estimated resistances of all of the legs, d) comparing thecalculated resistances of the legs between the boundary nodes with themeasured resistances of the legs between the boundary nodes, e) based onthe comparison, re-estimating the resistances of all of the legs; and f)iteratively repeating steps c)-e) until the measured resistances of thelegs between the boundary nodes converge to the calculated resistancesof the legs between the boundary nodes to thus accurately determine theresistances of the legs between or connected to the internal nodes. 33.An analysis method comprising: a) measuring the resistances of legs of agrid between boundary nodes of said grid; b) estimating the resistancesof all of the legs; c) calculating the resistances of all of the legsbased on the measured resistances of the legs between the boundary nodesand the estimated resistances of all of the legs; d) comparing thecalculated resistances of the legs between the boundary nodes with themeasured resistances of the legs between the boundary nodes; e) based onthe comparison, re-estimating the resistances of all of the legs; and f)iteratively repeating steps c)-e) until the measured resistances of thelegs between the boundary nodes converge to the calculated resistancesof the legs between the boundary nodes.
 34. An analysis methodcomprising: a) measuring a characteristic of legs of a grid betweenboundary nodes of said grid; b) estimating the same characteristic ofall of the legs; c) calculating the same characteristic of all of thelegs based on the measured characteristic of the legs between theboundary nodes and the estimated characteristic of all of the legs; d)comparing the calculated characteristic of the legs between the boundarynodes with the measured characteristic of the legs between the boundarynodes, e) based on the comparison, re-estimating the characteristic ofall of the legs; and f) iteratively repeating steps c)-e) until themeasured characteristic of the legs between the boundary nodes convergeto the calculated characteristic of the leg between the boundary node.35. The analysis method of claim 34 in which the characteristic is acomplex impedence.
 36. A sheet-like sensor for measuring stressdistribution comprising: a grid of members which change in resistancewhen subjected to strain, the members intersecting at internal nodes andintersecting at boundary nodes at the periphery of the grid defining aplurality of legs; and an analyzer electrically connected only to theboundary nodes and configured to calculate any change in resistance inall of the legs based solely on the measured resistance of the legsbetween the boundary nodes, said analyzer configured to: a) measure theresistances of the legs between the bound nodes, b) estimate theresistances of all of the legs, c) calculate the resistances of all ofthe legs based on the measured resistances of the legs between theboundary nodes and the estimated resistances of all of the legs, d)compare the calculated resistances of the legs between the boundarynodes with the measured resistances of the legs between the boundarynodes, e) based on the comparison, re-estimate the resistances of all ofthe legs, and f) iteratively repeat steps c)-e) until the measuredresistances of the legs between the boundary nodes converge to thecalculated resistances of the legs between the boundary nodes to thusaccurately determine the resistances of the legs between or connected tothe internal nodes.
 37. The sensor of claim 36 in which the analyzer isfurther configured to calculate the strain experience by each leg. 38.The sensor of claim 37 in which the analyzer is further configured toidentify any leg which has failed based on a very high determinedresistance.
 39. The sensor of claim 36 in which the estimate is based onthe step a).
 40. The sensor of claim 39 in which the estimate is set tothe mean of the measured resistances.
 41. The sensor of claim 36 inwhich relaxation is used in step f).
 42. A sheet-like sensor formeasuring stress distribution comprising: a grid of members which changein resistance when subjected to strain, the members intersecting atinternal nodes and intersecting at boundary nodes at the periphery ofthe grid defining a plurality of legs; and an analyzer connected only tothe boundary nodes, said analyzer configured to: a) measure theresistances of the legs between the boundary nodes, b) estimate theresistances of all of the legs, c) calculate the resistances of all ofthe legs based on the measured resistances of the legs between theboundary nodes and the estimated resistances of all of the legs, d)compare the calculated resistances of the legs between the boundarynodes with the measured resistances of the legs between the boundarynodes, e) based on the comparison, re-estimate the resistances of all ofthe legs, and f) iteratively repeat steps c)-e) until the measuredresistances of the legs between the boundary nodes converge to thecalculated resistances of the legs between the boundary nodes to thusaccurately determine the resistances of the legs between or connected tothe internal nodes.